The spectra of q-state vertex models and related antiferromagnetic quantum spin chains

Some exactly solvable q-state vertex models are investigated. The author employs inversion relations to determine directly the spectra of the transfer matrices in the thermodynamic limit by avoiding the more cumbersome Bethe Ansatz. The results are applied to related quantum spin chains of which two families are SU(q) and SO(q) invariant, respectively. Various quantities are calculated, e.g. energy-momentum excitations and the correlation length. For q=3 the SU(q) invariant chain is the pure biquadratic spin-1 Hamiltonian which turns out to be non-critical. The ground-state energy, the gap, and the correlation length are given.